# Derivative wrt one variable of an integral wrt another

• apb000
In summary, the conversation discusses the notation for finding the derivative of the integral of a function over a variable. The speaker clarifies that n is a function of w, and outside of the integral it is a variable. They also mention the relationship between functions and variables in the context of finding derivatives.
apb000
I can't seem to find this anywhere. What's
d/dn(∫ n(w) dw)?

That's the derivative wrt n of the integral of n over w (note: n is a function of w). Seems straightforward enough. Is there a simplification?

Thanks

The notation you are using to express what you want is confusing. Under the integral sign, n is a function. Outside it is a variable.

But a function is a variable! One can always define the derivative of y with respect to a function, f(x)- dy/dx= (dy/df)(df/dx) so dy/df= (dy/dx)/(df/dx).

So $d/df \int f(x)dx= (d(\int f(x)dx)/dx)(df/dx)= f(x)(dy/dx)$.

HallsofIvy said:
But a function is a variable! One can always define the derivative of y with respect to a function, f(x)- dy/dx= (dy/df)(df/dx) so dy/df= (dy/dx)/(df/dx).

So $d/df \int f(x)dx= (d(\int f(x)dx)/dx)(df/dx)= f(x)(dy/dx)$.
Where did y come from?

## What is the derivative with respect to one variable of an integral with respect to another?

The derivative with respect to one variable of an integral with respect to another is a mathematical concept that describes the rate of change of an integral with respect to one of its variables. It is calculated using the fundamental theorem of calculus, which states that the derivative of an integral is equal to the original function.

## How is the derivative with respect to one variable of an integral with respect to another calculated?

The derivative with respect to one variable of an integral with respect to another is calculated using the chain rule. This involves taking the derivative of the integrand with respect to the variable of interest, and then multiplying it by the derivative of the upper limit of integration with respect to that same variable.

## What is the significance of the derivative with respect to one variable of an integral with respect to another?

The derivative with respect to one variable of an integral with respect to another is important in many fields of science and mathematics, as it allows for the calculation of rates of change in complex systems. It is also used in optimization problems, where finding the minimum or maximum value of a function requires taking the derivative of an integral.

## Can the derivative with respect to one variable of an integral with respect to another be negative?

Yes, the derivative with respect to one variable of an integral with respect to another can be negative. This indicates that the integral is decreasing with respect to the variable of interest. However, it is also possible for the derivative to be zero or positive, indicating that the integral is either constant or increasing, respectively.

## Are there any practical applications of the derivative with respect to one variable of an integral with respect to another?

Yes, there are many practical applications of the derivative with respect to one variable of an integral with respect to another. For example, it is used in engineering to calculate rates of change in physical systems, in economics to model the change in demand or supply, and in physics to determine the velocity or acceleration of an object.

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